Vol. 10,No.1, January-February 1977
Polymer Chains Measured with Respect to Chain-Fixed Axes 139
Spans of Polymer Chains Measured with Respect to Chain-Fixed Axes Robert J. Rubin* and Jacob Mazur Institute for Materials Research, National Bureau of Standards, 20234. Received July 8,1976 Washington, D.C.
ABSTRACT: An N-step random walk on a cubic lattice is adopted as a model of a polymer chain. T h e span or extent of a random walk in a direction e is defined as the maximum distance between parallel planes normal to e which contain lattice points visited by that walk. T h e spans of each random walk configuration are measured with respect to two different sets of orthogonal axes determined by the configuration. T h e first set of orthogonal axes is based on the direction of the maximum span of the configuration. T h e second set is based on the directions of the principal components of the radius of gyration tensor of the chain Configuration. For each set of axes, a smallest right prism is determined whose edges are parallel to the chain-fixed axes and which contain all the steps of the random walk. A Monte Carlo procedure is used t o estimate the average largest, intermediate, and smallest spans, or prism dimensions. Both the simple unrestricted random walk with N = 50,100, 200 and the self-avoiding random walk with N = 50,75, 100, 150 are treated. In the case of the orthogonal axes based on the maximum span, the ratio of the average maximum span to the average smallest span is approximately independent of N and equal to 2.42 for the unrestricted walk and 2.73 for the self-avoiding walk. T h e distribution of steps inside the spanning right prisms is investigated by dissecting them in two different ways. First, the prism is cut in ten equal sections by a set of parallel, equally spaced planes which are normal to a n edge of the prism. T h e fraction of steps contained in pairs of sections which are equidistant from the central cutting plane is determined. This procedure is repeated in turn for each of the different edges of the prism. Second, a symmetric oval (the ellipsoid is a special case) is inscribed in the prism with its axes parallel to those of the prism. Four similar and successively smaller nested ovals are also introduced. For each random walk configuration, the fraction of steps contained in each oval shell is determined.
I. Introduction In this paper we continue our investigation of the spans of unrestricted and self-avoiding cubic-lattice-random-walk models of polymer chains.' The two terms, random walk and random polymer chain, are used interchangeably as are the terms step and polymer segment. The span of an N-step random walk in a direction e is defined as the maximum distance between parallel planes normal to e which contain lattice points visited by that walk. In the preceding paper,' RMI, the spans of each random walk are determined with respect to the directions of the principal axes of the cubic lattice, xl, x?, and x:+ The span in the x i direction is denoted by X , (N) and the values of these spans are ordered according to their magnitude. The ordered spans are denoted by the symbol & ( N )where {:i(N) 2 & ( N )b .$1(N).These three spans, ( N ) , i = 1 , 2 , 3 ,define the dimensions of the smallest right prism, II,(N), whose edges are parallel to the x i axes and which contains all the steps in the random walk. One of the principal results obtained in RMI is that the right prism whose edges have the lengths (E:j(N)), ( . $ z ( N ) and ) , ((1(N)) is significantly noncubic where (ti( N ) )denotes the average value of E, ( N ) . The relative proportions, ( ~ ~ ( N ) ) : ( . $ z ( N ) ) :in( ~the l (limit N)) N m are 1.637:1.267:1 in the case of the unrestricted random walk and 1.75:1.31:1 in the case of the self-avoiding random walk. This result, that the typical or average shape of a polymer chain is asymmetric, is not new. Kuhn2 presented approximate arguments leading to a similar conclusion over 40 years ago. H o l l i n g ~ w o r t hand ~ ~ later ~ Weidmann, Kuhn, and Kuhns elaborated on these same arguments. More recently, K ~ y a m a ,Sol5 ~ , ~ and Stockmayer,8 and SolEg used Monte Carlo methods to investigate the principal components of the moment of inertia tensor, or radius of gyration tensor, of an unrestricted random walk. SolE and Stockmayer found that the relative values of the average ordered principal components were 11.69:2.69:1 for N = 50 and 100. Mazur, Guttman, and McCrackin'O have extended these calculations to the case of self-avoiding random walks and have obtained for the corresponding relative proportions 14.81:3.07:1. In the dilute solution properties of random chain polymer molecules, there
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* National Institutes of Health, Bethesda, Md. 20014.
are many immediate and obvious physical consequences of the asymmetric dimensions of the average spanning prism of the typical molecule. Proper account of the asymmetry must be taken in all experiments involving motion of the entire chain or parts of it, relative to the solvent such as experiments on viscosity, streaming birefringence, dielectric relaxation, and rates of sedimentation and diffusion. The average dimensions of the spanning prism also figure in theoretical models of peak migration in gel permeation chromatography. The spans in RMI are measured with respect to the spacefixed lattice directions, XI,x2, x3. In the present paper, we investigate the spans with respect to directions determined by the random walk configuration itself. In this way we can obtain more explicit and detailed information regarding the asymmetric shape of polymer chains. Two different sets of orthogonal chain-fixed directions are considered. The first set, based on the direction of the maximum span of the random walk, is defined as follows. Consider the set of all arbitrarily oriented rectangular prisms with minimum dimensions which contain the random walk. Determine the subset of prisms which has the longest edge. The edge length, R3(N),is the maximum span of the random walk configuration. Then for this subset determine the prism(s), I I R ( N ) ,with the next longest edge and denote this edge length by R2(N).The third edge length, R1(N), is the smallest span associated with the direction of the maximum span R3(N) and Rp(N).Note that the direction associated with the maximum span may not be unique. That is, two different pairs of steps in the random walk configuration might be separated by the same distance, Ry(N).Note also that the smallest span R1(N), which is associated with R x ( N ) ,is not necessarily the minimum span of the random walk configuration. The second set of orthogonal chain-fixed directions which we consider is the set associated with the directions of the principal axes of inertia of the chain. First, the principal components of the radius of gyration and the directions of the principal components are determined for the chain. Then the principal components of the square radius of gyration are ordered according to their magnitude. The j t h ordered component of the square radius of gyration is denoted by S , 2 ( N ) ,j = 1, 2, 3, and S&N) 2 S22(N)2 S12(N).The
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Macromolecules rections. Second, we have introduced a different type of subdivision of the spanning prism. A symmetric oval (the ellipsoid is a special case) is inscribed in the prism with its axes parallel to those of the prism. Four similar and successively smaller nested ovals are also introduced. For each random walk configuration, the fraction of segments contained in each oval shell is determined. The foregoing calculations of a minimum-size right prism which contains all segments of a given random walk as well as the calculation of the distribution of segments within the right prism have been carried out for both unrestricted randomwalk and self-avoiding-random-walk models of polymer chains. As in RMI, the self-avoiding walks are generated by using the method of Rosenbluth and Rosenbluthll which is also described in detail in papers by Mazur and McCrackin12 and McCrackin, Mazur, and Guttman.13 In this method, the estimated sample average of a physical parameter generally differs from the true average. This difference, or bias, is negligible in a sufficiently large sample.14J5The magnitude of the bias in our calculations is discussed in Appendix A, using the average maximum span (R3(N)) as an example.
2. Spans Associated with t h e Maximum Span
Figure 1. (A) The two sets of orthogonal axes ( X I , ZZ. X Z ] and { x I’, X Z ’ , x:!’J are shown as well as the angles CY and 13. (B) The two sets of orthogonal axes 1x1, X Z , x ~and J (XI”, x p ” , xgl’l are shown as well as the angles a , p, and y.
span, r j ( N ) ,in the direction of the j t h ordered component, S j z ( N ) is , determined for each principal direction. In this way, a smallest rectangular prism, II,(N), whose edges are parallel to the directions of the principal axes of inertia is determined for each random walk configuration. In section 2 we describe a Monte Carlo procedure for calculating estimates of the first and second moments of the spans Ri(N), i = 1, 2, 3, of unrestricted and self-avoiding walks. Estimated values of the first and second moments are obtained for the unrestricted random walk ( N = 50,100,200) and the self-avoiding walk ( N = 50, 75,100,150). In section 3 we describe an analogous procedure for the calculation of the first and second moments of the spans r i ( N ) ,i = 1 , 2 , 3, associated with the ordered principal components of the square radius of gyration. In this case, estimated values of the moments are presented for unrestricted ( N = 50,100,200)and self-avoiding ( N = 50,75,100, 150) walks. In addition to determining the ordered dimensions of a minimum-size right prism, II,(N) or IIR(N),which contains all polymer chain segments, we investigate the distribution of segments inside these right prisms by dissecting them in two different ways. First, the prism is cut in ten sections by a set of parallel, equally spaced planes which are normal to an edge of the prism. The number of polymer segments contained in the pairs of sections which are equidistant from the central cutting plane is determined. This procedure is repeated in turn for each of the different edges of the prism. The three segment density distributions which are obtained in this way for the case of the prisms, II,(N), can be used to reconcile an apparent discrepancy between the relative values obtained for ( S , 2 ( N ) )j, = 1,2,3, by K0yama,63~Sol5 and Stockmayer? SolE? and Mazur, Guttman, and McCrackinlO and the relative values of the squares of the spans in the corresponding di-
Consider the set of spans measured with respect to chainfixed axes which is based on the maximum span. In an N-step random walk which starts at the origin of a simple cubic lattice, the coordinates of the j t h step are denoted by the vector xG) whose components are x l G ) , x z G ) , x 3 G ) with 0 < j < N and xl(0) = x z ( O ) = xa(0) = 0. The maximum span, Ra(N),is the maximum distance in the set of ‘/2N(N 1) inter-step distances,
+
The direction associated with Ra(N) is the direction of the vector xG’)- x(k’) connecting the pair of maximally separated steps. If there is not a unique maximum, that is, if more than one pair of steps is associated with R3(N), then the direction associated with one of the pairs is arbitrarily selected. To determine the maximal spans of the random walk in directions orthogonal to xG’)- x(k’), we express the xlG), x z G ) , x C 3 G coordinates ) in a new coordinate system xl’, xz’, xg’ where the positive x 3 ’ direction coincides with the direction of xG’) - x ( k ’ ) . The transformation of coordinates, specified in terms of the two angles CY and p depicted in Figure 1,has the form cos CY -sin CY cos p sin CY sin p
sin CY cos CY cos p -cos CY sin p
sin /3 cos /3
where x , ’ G ) is the coordinate of particle j in the xL’ direction. The maximum span in a direction transverse to xG’)- x(k’), R2(N),is the maximum transverse distance in the set of %N(N 1)distances
+
([xI’G) - x1’(k)l2 + [ x ~ ’ G-) ~ z ’ ( k ) ] ” ’ ’ ~ O
